If the Move, Copy, Rotate commands don't fit your situation, another option would be to sketch a construction line down the center of your sketch, use it to mirror the original sketch entites, then go back and delete them.

SW does a very poor job of allowing users to control front and back sides of planes. This has been an issue in the past.

One way is to change the way the plane is defined. Start with a 3D sketch that has a line that defines the plane normal and has an endpoint at the desired plane location. Define a plane using the line endpoint and the line segment as references. To turn the plane around, edit the sketch and rotate the line about the defining endpoint 180 degrees.